This simple, elegant construction yields a square simply by using the steps to bisect a segment.  Begin by using your compass to draw a circle, making sure to mark its center.  With your straight edge, draw the diameter of the circle.  This diameter is the segment you will bisect, which will produce a new segment perpendicular to the diameter.  Extend that segment so that it touches the circle, and you now have four evenly spaced points on your circle.  Connect them to make a square.

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Perform the same steps to bisect all four sides of the square, and you will have the points you need to make an octagon!

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